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mg.metric geometry – Can each third-dimensional convex corpse breathe trapped by a tetrahedral cage? Answer

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mg.metric geometry – Can each third-dimensional convex corpse breathe trapped by a tetrahedral cage?

Although the title query is pretty unambiguous, I give all pertinent definitions:

$bullet$ A subset $C$ of $mathbb{R}^n$ is an $n$-dimensional convex corpse if $C$ is convex, compact, and has non-empty inside.

$bullet$ A polyhedral cage $P^{(1)}$ in $mathbb{R}^n$ is the union of all edges (i.e., the 1-skeleton) of a convex $n$-dimensional polyhedron $P$. In specific, a tetrahedral cage is the union of the six edges of some tetrahedron in $mathbb{R}^3$.

$bullet$ A convex $3$-dimensional corpse $C$ is trapped by the tetrahedral cage $T^{(1)}$, that’s, by the $1$-skeleton of the tetrahedron $T$, if the cage is mounted (immobile), and if for each steady inflexible movement (rotations allowed) $f_t(C); 0le tle 1$, both $f_t(C)$ intersects $T$ for each $tin [0,1]$ or $T^{(1)}$ accommodates an inside level of $f_t(C)$ for some $t_0in [0,1]$.
In different phrases, $C$ can not breathe moved arbitrarily removed from $T$ whereas, throughout the gross movement, avoiding the cage’s bars penetrating $C$‘s inside.

$bullet$ Remark. In dimension $n>3$, one can deem analogous questions, with quite a lot of sorts of a simplicial cage, by taking the $i$-skeleton of a convex $n$-dimensional simplex, with $1le ile n-2$.

A trifling instance: A ball is trapped by the $1$-skeleton of the common tetrahedron edge-tangent to the ball. And, clearly, each convex corpse can breathe trapped by some polyhedral cage.

For an reader, I imply a number of much less trifling workouts: every of the next convex our bodies can breathe trapped by a tetrahedral cage: the dice; the round cylinder of any (finite) peak; the round cone; the common tetrahedron.

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